Question: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-2x+5y &= 6 \\ -2x+9y &= 8\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $9y = 2x+8$ Divide both sides by $9$ to isolate $y$ $y = {\dfrac{2}{9}x + \dfrac{8}{9}}$ Substitute this expression for $y$ in the first equation. $-2x+5({\dfrac{2}{9}x + \dfrac{8}{9}}) = 6$ $-2x + \dfrac{10}{9}x + \dfrac{40}{9} = 6$ Simplify by combining terms, then solve for $x$ $-\dfrac{8}{9}x + \dfrac{40}{9} = 6$ $-\dfrac{8}{9}x = \dfrac{14}{9}$ $x = -\dfrac{7}{4}$ Substitute $-\dfrac{7}{4}$ for $x$ back into the top equation. $-2( -\dfrac{7}{4})+5y = 6$ $\dfrac{7}{2}+5y = 6$ $5y = \dfrac{5}{2}$ $y = \dfrac{1}{2}$ The solution is $\enspace x = -\dfrac{7}{4}, \enspace y = \dfrac{1}{2}$.